Please Login to access more options.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |
We'll be finishing up Darrel's proof that a polynomial of degree $n$ can have at most $n$ zeros, counting multiplicity. We'll also have Tara share her example of a PID such that the polynomial ring is not a PID. Brennan's excited to share his work on last Wednesday's #4. Then we'll work on as many of the following as there is time.
We'd like to know when we obtain polynomials $p(x)\in F[x]$ such that $F[x]/\left<p(x)\right>$ is a field. We already know this means that $I=\left<p(x)\right>$ must be a maximal ideal in $F[x]$. We've also seen that if we can factor $p(x)$ as the product of two polynomials (that don't have multiplicative inverses), then $I$ is not maximal. Let's make a definition to make this precise.
Definition (Irreducible Polynomial Reducible Polynomial)
Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product $f(x)=g(x)h(x)$, with $g(x)$ and $h(x)$ from $D[x]$, then $g(x)$ or $h(x)$ is a unit in $D[x]$. A nonzero, nonunit element of $D[x]$ that is not irreducible over $D$ is called reducible over $D$.
In general, it's a hard problem to determine when a polynomial is reducible or irreducible. However, there are a few cases where it's easy.
Problem 53 (Reducibility Test For Degrees 2 And 3)
Let $F$ be a field. If $f(x)\in F[x]$ and deg $f$ is 2 or 3, then prove that $f(x)$ is reducible over $F$ if and only if $f(x)$ has a zero in $F$.
Before we look at any other reducibility tests, let's prove a key theorem, namely that $p(x)$ is irreducible if and only if $\left<p(x)\right>$ is maximal. We really want to know when we can guarantee that $F[x]/\left<p(x)\right>$ is a field.
Problem 54 (We Know $\left<p(x) \right>$ Is Maximal Iff $p(x)$ Is Irreducible)
Let $F$ be a field and let $p(x)\in F[x]$. Prove that $\left<p(x) \right>$ is a maximal ideal if and only if $p(x)$ is irreducible over $F$.
Now let's look at some other tests for reducibility. This next definition just gets rid of a complication from some polynomials, by removing from the polynomial any common factors.
Definition (Content Of A Polynomial Primitive Polynomial)
The content of a polynomial in $\mathbb{Z}[x]$ is the greatest common divisor of the coefficients. A primitive polynomial has content 1.
Problem 54.5 (The Product Of Primitives Is Primitive)
Prove that the product of two primitive polynomials is primitive.
As you work through the problems in the next few weeks, you'll want to pay close attention to the assumptions. In some problems we assume that we are working in a field. In some problems, we assume that we are working in an integral domain. The next problem shows that if you can show something is reducible over the field $\mathbb{Q}$, then it is reducible over $\mathbb{Z}$.
Problem 55 (Reducibility Over Q Implies Reducibility Over Z)
Let $f(x)\in \mathbb{Z}[x]$. Prove that if $f(x)$ is reducible over $\mathbb{Q}$, then $f(x)$ is reducible over $\mathbb{Z}$.
The contrapositive to the previous problem is extremely powerful, namely if a polynomial with integer coefficients is not reducible over $\mathbb{Z}$, then it is not reducible over $\mathbb{Q}$. For this reason, we'll now study irreducibility tests over $\mathbb{Z}$.
Problem 56 (Mod P Irreducibility Test)
Let $p$ be a prime and suppose that $f(x)\in \mathbb{Z}[x]$. Let $\bar f (x)$ be the polynomial in $\mathbb{Z}_p[x]$ obtained by reducing the coefficients of $f(x)$ modulo $p$. Prove that if if $\bar f (x)$ is irreducible over $\mathbb{Z}_p$ and $\text{deg }\bar f(x) = \text{deg }f(x)$, then $f(x)$ is irreducible over $\mathbb{Q}$.
Problem 57 (Eisenstein's Criterion)
Let $f(x)=a_nx^n + \cdots +a_1x +a_0$. Prove that if there is a prime $p$ such that $p$ divides every coefficient but $a_n$ and $p^2$ does not divide $a_0$, then $f(x)$ is irreducible over $\mathbb{Q}$.
For more problems, see AllProblems